Dynamical complexities in a predator-prey system involving teams of two prey and one predator

Abstract

Empirical studies have shown that animals often focus on short-term benefits under conditions of predation risk, which increases the likelihood that they will co-operate with others. With this motivation, we propose and analyze a predator and two prey model with the assumption that during predation the members of both prey make a team to reduce risk of predation. We incorporate Monod–Haldane and Holling type II functional response to model the interaction with predator. Firstly, we discuss conditions which ensure that model system has a unique positive solution. We investigate stability and Hopf bifurcation conditions to explore dynamics of system around positive equilibrium. We also derive Kolmogorov conditions for the parametric restriction of the system. Secondly, we present numerical solution which substantiate our analytical results. In numerical simulation, we observe period doubling and period halving cascade which explore the dynamical complexity of predator-prey system. Finally, we conclude that partial co-operation and low defense may lead to extinction of prey species.

Mathematics Subject Classification

Notes

Acknowledgements

The work done in this paper is supported by a grant (File No. ECR/2017/000141) under Early Career Research (ECR) Award, Science and Engineering Research Board (SERB), Department of Science and Technology, New Delhi, to the corresponding author (SNR).

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